Question

A chord of a circle has a length of 12 cm. The angle subtended by the chord at a point on the circumference is 30°. What is the distance from the center of the circle to the chord?

• A. 6√3 cm
• B. 3√3 cm
• C. 6 cm
• D. 3 cm

Answer 1

Answer:

A. 6√3 cm

Given:

• Chord length (=12) cm

• Angle subtended by the chord at the circumference (=30^\circ)

The angle subtended by the same chord at the centre is twice the angle at the circumference:

$\angle AOB = 2 \times 30^\circ = 60^\circ.$

Let the radius be (r).

Using the chord-length formula:

$\text{Chord} = 2r\sin\left(\frac{\theta}{2}\right)$

where (\theta=60^\circ).

$12 = 2r\sin30^\circ$
$12 = 2r\left(\frac12\right)=r$

So, the radius is

$r=12\text{ cm}.$

The perpendicular distance (d) from the centre to the chord is

$d=r\cos\left(\frac{\theta}{2}\right)$
$=12\cos30^\circ$
$=12\cdot\frac{\sqrt3}{2}$
$=6\sqrt3\text{ cm}.$

Answer:

$\boxed{6\sqrt3\ \text{cm}}$